The study of nonlinear Alfvén waves arising in a finite-beta plasma leads to the following Cauchy problem: \[ u_t+ iu_{xx}= \alpha(u|u|^2)_x+ \beta(u H(|u|^2))_x,\;u(x, 0)= \varphi(x), \] \(\alpha\), \(\beta\) are real constants, \(H(\cdot)\) denotes the Hilbert transform. Using global a priori estimates, the authors establish global existence and uniqueness of smooth solutions and discuss their asymptotic behaviour as \(\alpha, \beta\to 0\).